396] TRIANGLE INSCRIBED IN A CIRCLE. 81 
then x, y, z will be as 
(b - c) {(2M— be) u — av\ 
: (c — a) {(2M— ca) u — bv] 
: (a — b) {(21/ — ab)u — cv], 
and substituting these values in the equation 
ax (y ~ + by (z — x) 2 + cz (sc — y) 2 = 0 
of the cubic, we have a cubic equation for the ratio (u : v); and thence the values 
(x, y, z) for the coordinates of the inflexions. 
It may be added, that we have 
12MU = - 3 (<aX 2 + bY> + cZ 2 ) {{be + M) x+(ca+M) y + (ab + M)z] 
+ {X 2 {I'Y - F'Z) + Y 2 {J'Z - G'X) + X 2 (K'X - H'Y) = 0, 
which is the equation of the cubic expressed in the canonical form. 
Pp. 175—179. Effecting the process indicated p. 73, but writing for greater con 
venience (x, y, z) in place of (X, F, Z), so that the substitution to be made is 
(a , b, c, f, g, h, i , j , k, l ) 
= {6x, Qy, 6z, —2y, —2z, —2x, —2z, —2x, —2y, x + y + z), 
respectively (where I have corrected a misprint in the formula as originally given) I 
find the equation of the envelope to be 
4yz (y - z)~ a 4 
+ 4zx (z — xf b 4 
+ 4>xy (x — y)~ c 4 
+ 4zx (z 2 + x 2 + 3yz — 2zx + 5xy) b 3 c 
+ 4 xy (x~ + y 2 + 3zx — 2 xy + 5 yz) c 3 a 
+ 4yz (y 2 + z 2 + 3xy — 2yz + 5zx) a 3 b 
-f 4xy (a? + y 2 + 3yz - 2xy -t- 5zx) be 3 
+ 4yz (y 2 + z 2 + 3zx - 2yz + 5xy) ca 3 
+ 4zx (z 2 +x 2 + 3xy - 2zx + byz) ab 3 
+ x (a? - 2x 2 y - 2a?z + xy 2 + 38xyz + xz 2 + 12y 2 z + 12y^ 2 ) b 2 c 2 
+ y (y3 _ 2y>z - 2y 2 x + yz 2 + 38xyz + yx 2 + 12z 2 x + 12zx 2 ) c 2 a 2 
+ z (z 3 - 2z 2 x - 2z 2 y + zx 2 + 38xyz + zy 2 + 12x 2 y + 12xy 2 ) a 2 b 2 
+ 2yz (11 a? + y 2 + z 2 - 2yz + 24>xy + 24zx) a 2 bc 
+ 2zx (lly 2 + z 2 + x 2 - 2zx + 24yz + 24xy) b 2 ca 
+ 2xy (11 z 2 + x 2 + y 2 - 2xy + 24^ + 24yz) c 2 ab = 0. 
C. VI. 
11